Presentation on theme: "Theoretical Physics Course codes: Phys2325"— Presentation transcript:
1Theoretical Physics Course codes: Phys2325 Course Homepage:Lecturer: Z.D.Wang,Office: Rm528, Physics BuildingTel:Student Consultation hours: 2:30-4:30pm TuesdayTutor: Miss Liu Jia, Rm525
2Text Book: Lecture Notes Selected from Mathematical Methods for PhysicistsInternational Edition (4th or 5th or 6th Edition) byGeorge B. Arfken and Hans J. WeberMain Contents:Application of complex variables,e.g. Cauchy's integral formula, calculus of residues.Partial differential equations.Properties of special functions(e.g. Gamma functions, Bessel functions, etc.).Fourier Series.Assessment:One 3-hour written examination (80% weighting)and course assessment (20% weighting)
31 Functions of A Complex Variables I Functions of a complex variable provide us some powerful andwidely useful tools in in theoretical physics.• Some important physical quantities are complex variables (thewave-function )• Evaluating definite integrals.• Obtaining asymptotic solutions of differentialsequations.• Integral transforms• Many Physical quantities that were originally real become complexas simple theory is made more general. The energy( the finite life time).
41.1 Complex Algebra We here go through the complex algebra briefly. A complex number z = (x,y) = x + iy, Where.We will see that the ordering of two real numbers (x,y) is significant,i.e. in general x + iy y + ixX: the real part, labeled by Re(z); y: the imaginary part, labeled by Im(z)Three frequently used representations:(1) Cartesian representation: x+iy(2) polar representation, we may writez=r(cos + i sin) orr – the modulus or magnitude of z- the argument or phase of z
5r – the modulus or magnitude of z - the argument or phase of zThe relation between Cartesianand polar representation:The choice of polar representation or Cartesian representation is amatter of convenience. Addition and subtraction of complex variablesare easier in the Cartesian representation. Multiplication, division,powers, roots are easier to handle in polar form,
6From z, complex functions f(z) may be constructed. They can be written Using the polar form,From z, complex functions f(z) may be constructed. They can be writtenf(z) = u(x,y) + iv(x,y)in which v and u are real functions.For example if , we haveThe relationship between z and f(z) is best pictured as a mapping operation, we address it in detail later.
7Function: Mapping operation vyZ-planeuxThe function w(x,y)=u(x,y)+iv(x,y) maps points in the xy plane into pointsin the uv plane.SinceWe get a not so obvious formula
8Special features: single-valued function of a Complex Conjugation: replacing i by –i, which is denoted by (*),We then haveHenceNote:ln z is a multi-valued function. To avoid ambiguity, we usually set n=0and limit the phase to an interval of length of 2. The value of lnz withn=0 is called the principal value of lnz.Special features: single-valued function of areal variable multi-valued function
101.2 Cauchy – Riemann conditions Having established complex functions, we now proceed todifferentiate them. The derivative of f(z), like that of a real function, isdefined byprovided that the limit is independent of the particular approach to thepoint z. For real variable, we require thatNow, with z (or zo) some point in a plane, our requirement that thelimit be independent of the direction of approach is very restrictive.Consider,
11Assuming the partial derivatives exist. For a second approach, we set Let us take limit by the two different approaches as in the figure. First,with y = 0, we let x0,Assuming the partial derivatives exist. For a second approach, we setx = 0 and then let y 0. This leads toIf we have a derivative, the above two results must be identical. So,,
12These are the famous Cauchy-Riemann conditions. These Cauchy- Riemann conditions are necessary for the existence of a derivative, thatis, if exists, the C-R conditions must hold.Conversely, if the C-R conditions are satisfied and the partialderivatives of u(x,y) and v(x,y) are continuous, exists. (see the proofin the text book).
13Analytic functionsIf f(z) is differentiable at and in some small region around ,we say that f(z) is analytic atDifferentiable: Cauthy-Riemann conditions are satisfiedthe partial derivatives of u and v are continuousAnalytic function:Property 1:Property 2: established a relation between u and vExample:
141.3 Cauchy’s integral Theorem We now turn to integration.in close analogy to the integral of a real functionThe contour is divided into n intervals .Letwiith for j. ThenThe right-hand side of the above equation is called the contour (path) integral of f(z)
15As an alternative, the contour may be defined by with the path C specified. This reduces the complex integral to thecomplex sum of real integrals. It’s somewhat analogous to the case ofthe vector integral.An important examplewhere C is a circle of radius r>0 around the origin z=0 in the direction of counterclockwise.
16In polar coordinates, we parameterize and , and havewhich is independent of r.Cauchy’s integral theoremIf a function f(z) is analytical (therefore single-valued) [and its partial derivatives are continuous] through some simply connected region R, for every closed path C in R,
17Stokes’s theorem proof Proof: (under relatively restrictive condition: the partial derivative of u, vare continuous, which are actually not required but usuallysatisfied in physical problems)These two line integrals can be converted to surface integrals byStokes’s theoremUsing andWe have
18For the real part, If we let u = Ax, and v = -Ay, then = [since C-R conditions ]For the imaginary part, setting u = Ay and v = Ax, we haveAs for a proof without using the continuity condition, see the text book.The consequence of the theorem is that for analytic functions the lineintegral is a function only of its end points, independent of the path ofintegration,
19•Multiply connected regions The original statement of our theorem demanded a simply connectedregion. This restriction may easily be relaxed by the creation of abarrier, a contour line. Consider the multiply connected region ofFig.1.6 In which f(z) is not defined for the interior RCauchy’s integral theorem is not valid for the contour C, but we canconstruct a C for which the theorem holds. If line segments DE andGA arbitrarily close together, then1.6 Fig.
211.4 Cauchy’s Integral Formula Cauchy’s integral formula: If f(z) is analytic on and within a closed contour C thenin which z0 is some point in the interior region bounded by C. Note thathere z-z0 0 and the integral is well defined.Although f(z) is assumed analytic, the integrand (f(z)/z-z0) is notanalytic at z=z0 unless f(z0)=0. If the contour is deformed as in Fig.1.8Cauchy’s integral theorem applies.So we have
22Let , here r is small and will eventually be made to approach zero(r0)Here is a remarkable result. The value of an analytic function is given atan interior point at z=z0 once the values on the boundary C arespecified.What happens if z0 is exterior to C?In this case the entire integral is analytic on and within C, so theintegral vanishes.
23DerivativesCauchy’s integral formula may be used to obtain an expression forthe derivation of f(z)Moreover, for the n-th order of derivative
24We now see that, the requirement that f(z) be analytic not only guarantees a first derivative but derivatives of all orders as well! The derivatives of f(z) are automatically analytic. Here, it is worth to indicate that the converse of Cauchy’s integral theorem holds as wellMorera’s theorem:
262.In the above case, on a circle of radius r about the origin, then (Cauchy’s inequality)Proof:where3. Liouville’s theorem: If f(z) is analytic and bounded in the complexplane, it is a constant.Proof: For any z0, construct a circle of radius R around z0,
27Since R is arbitrary, let , we have Conversely, the slightest deviation of an analytic function from aconstant value implies that there must be at least one singularitysomewhere in the infinite complex plane. Apart from the trivial constantfunctions, then, singularities are a fact of life, and we must learn to livewith them, and to use them further.
281.5 Laurent Expansion Taylor Expansion Suppose we are trying to expand f(z) about z=z0, i.e.,and we have z=z1 as the nearest point for which f(z) is not analytic. Weconstruct a circle C centered at z=z0 with radiusFrom the Cauchy integral formula,
29Here z is a point on C and z is any point interior to C Here z is a point on C and z is any point interior to C. For |t| <1, wenote the identitySo we may writewhich is our desired Taylor expansion, just as for real variable powerseries, this expansion is unique for a given z0.
30Schwarz reflection principle From the binomial expansion of for integer n (as anassignment), it is easy to see, for real x0Schwarz reflection principle:If a function f(z) is (1) analytic over some region including the real axisand (2) real when z is real, thenWe expand f(z) about some point (nonsingular) point x0 on the real axisbecause f(z) is analytic at z=x0.Since f(z) is real when z is real, the n-th derivate must be real.
31Laurent SeriesWe frequently encounter functions that are analytic in annular region
32Drawing an imaginary contour line to convert our region into a simply connected region, we apply Cauchy’s integral formula for C2 and C1,with radii r2 and r1, and obtainWe let r2 r and r1 R, so for C1, while for C2,We expand two denominators as we did before(Laurent Series)
33whereHere C may be any contour with the annular region r < |z-z0| < R encircling z0 once in a counterclockwise sense.Laurent Series need not to come from evaluation of contour integrals. Other techniques such as ordinary series expansion may provide the coefficients.Numerous examples of Laurent series appear in the next chapter.
34Example:(1) Find Taylor expansion ln(1+z) at point z(2) find Laurent series of the functionIf we employ the polar formThe Laurent expansion becomes
35For example Analytic continuation which has a simple pole at z = -1 and is analyticelsewhere. For |z| < 1, the geometric series expansion f1, while expanding it about z=i leads to f2,
36Suppose we expand it about z = i, so that converges for (Fig.1.10)The above three equations are different representations of the samefunction. Each representation has its own domain of convergence.A beautiful theory:If two analytic functions coincide in any region, such as the overlap of s1 and s2,of coincide on any line segment, they are the same function in the sense that theywill coincide everywhere as long as they are well-defined.